The lifespans of sloths in a particular zoo are normally distributed. The average sloth lives $17$ years; the standard deviation is $2.5$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a sloth living longer than $14.5$ years.
Solution: $17$ $14.5$ $19.5$ $12$ $22$ $9.5$ $24.5$ $68\%$ $16\%$ $16\%$ We know the lifespans are normally distributed with an average lifespan of $17$ years. We know the standard deviation is $2.5$ years, so one standard deviation below the mean is $14.5$ years and one standard deviation above the mean is $19.5$ years. Two standard deviations below the mean is $12$ years and two standard deviations above the mean is $22$ years. Three standard deviations below the mean is $9.5$ years and three standard deviations above the mean is $24.5$ years. We are interested in the probability of a sloth living longer than $14.5$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $68\%$ of the sloths will have lifespans within 1 standard deviation of the average lifespan. The remaining $32\%$ of the sloths will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({16\%})$ will live less than $14.5$ years and the other half $({16\%})$ will live longer than $19.5$ years. The probability of a particular sloth living longer than $14.5$ years is ${68\%} + {16\%}$, or $84\%$.